A lucky factorization

Factorizations, using the post-processing program Msieve, usually take an average of 2 attempted sqrt calculations to split composite numbers into two factors. Sometimes we are unlucky and it takes three, four or more times. If there are more factors, more sqrts are required.

In this case it took 33. Yes, that’s right 33. The mathematical probability is 1/2 for each step to split the number into two factors. Here, things went wrong when almost every step said “Newton iteration failed to converge”, so it seems there are hidden bugs there somewhere.

The sqrt phase started on Saturday morning (Oct 29, 2011) and continued until Monday morning (Oct 31).

The Lanczos matrix step said that it had found 40 nontrivial dependencies, so if all 40 had failed, we would normally either reduce the number of relations from the 73,520,074 found or continue the sieving to find more. Either way would involve another possible 21+ hours of matrix inversion followed by the sqrt steps.

What a relief.

By the way, the number being factored was the 220-digit composite from 13148 + 48131 = 7 • C220.